On the integrability of spatial polygons in 3 dimensional semi Riemann spaces

Integrability of the Pentagram Map

The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresp…

Period matrices of some hyperelliptic Riemann surfaces

In this paper, we calculate period matrices of algebraic curves defined by $$w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2)\cdots (z^2-a_{g-1}^2)$$ for any $g\geq 2$ and $a_1, a_2, \dots, a_{g-1}\in \mathbb{R}$ with $1<a_1<a_2<\cdots <a_{g-1}$. We construct these algebraic curves from Euclidean polygons. A symplectic basis of these curves are given from the polygons.

The Algebraic Dynamics of the Pentagram Map

The pentagram map, introduced by Schwartz in 1992, is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev's proof of complex integrability. In the course of the proof, w…

跨越超越性鸿沟：从平移曲面到代数曲线

Crossing the transcendental divide: from translation surfaces to algebraic curves

本文利用离散 Riemann 曲面理论提出算法，从可方格化的平移曲面逼近 Jacobian 簇，并在 L 形多边形与 Jenkins-Strebel 代表上验证，结合 Riemann theta 函数给出至亏格 5 的数值实验和猜想。